# On Upper Total Domination Versus Upper Domination in Graphs

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## Abstract

A total dominating set of a graph *G* is a dominating set *S* of *G* such that the subgraph induced by *S* contains no isolated vertex, where a dominating set of *G* is a set of vertices of *G* such that each vertex in \(V(G){\setminus } S\) has a neighbor in *S*. A (total) dominating set *S* is said to be minimal if \(S{\setminus } \{v\}\) is not a (total) dominating set for every \(v\in S\). The upper total domination number \(\varGamma _t(G)\) and the upper domination number \(\varGamma (G)\) are the maximum cardinalities of a minimal total dominating set and a minimal dominating set of *G*, respectively. For every graph *G* without isolated vertices, it is known that \(\varGamma _t(G)\le 2\varGamma (G)\). The case in which \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\) has been studied in Cyman et al. (Graphs Comb 34:261–276, 2018), which focused on the characterization of the connected cubic graphs and proposed one problem to be solved and two questions to be answered in terms of the value of \(\frac{\varGamma _t(G)}{\varGamma (G)}\). In this paper, we solve this problem, i.e., the characterization of the subcubic graphs *G* that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\), by constructing a class of subcubic graphs, which we call triangle-trees. Moreover, we show that the answers to the two questions are negative by constructing connected cubic graphs *G* that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}>\frac{3}{2}\) and a class of regular non-complete graphs *G* that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\).

## Keywords

Upper domination number Upper total domination number Subcubic graphs## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (61872101, 61672051, 61702075), the China Postdoctoral Science Foundation under Grant (2017M611223).

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